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Chapter 4: Problem 530

Graph the linear inequality \(3 x+2 y \geq-6\)

Short Answer

Expert verified
Graph the line y = -3/2x - 3 and shade above it.

Step by step solution

01

Rewrite the inequality in slope-intercept form

To graph the inequality, first rewrite it in the form \(y = mx + b\). Start by isolating y: \(3x + 2y \geq -6\).Subtract 3x from both sides: \(2y \geq -3x -6\). Divide everything by 2: \(y \geq -\frac{3}{2}x - 3\).
02

Graph the line

Graph the line \(y = -\frac{3}{2}x - 3\). Use a solid line because the inequality is \(\geq\), meaning points on the line are included in the solution. Start at the y-intercept (0, -3), then use the slope \(-\frac{3}{2}\) to plot another point: go down 3 units and right 2 units.
03

Determine the shading region

Since the inequality is \(y \geq -\frac{3}{2}x - 3\), shade the region above the line. Choose a point not on the line, such as (0, 0), to test the inequality. If the inequality holds when substituting this point, then shade that side.
04

Test a point

Test the point (0, 0): \(0 \geq -\frac{3}{2}(0) - 3\), which simplifies to \(0 \geq -3\).This is true, so shade the region above the line, including the line itself.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is an easy way to write the equation of a line. It looks like this: \[ y = mx + b \] Here, m is the slope, and b is the y-intercept where the line crosses the y-axis.
To rewrite the inequality from 3x + 2y \geq -6 into slope-intercept form, follow these steps:
  • Isolate y by subtracting 3x from both sides to get 2y \geq -3x -6.
  • Divide every term by 2 to solve for y: \[ y \geq -\frac{3}{2}x - 3 \]

The inequality is now in the familiar slope-intercept form y = mx + b, ready for graphing!
Shading Regions
Shading regions on a graph shows the solution set of an inequality. For y \geq -\frac{3}{2}x - 3, you need to determine where to shade. The rule is:
  • Greater than or equal to (\( \geq \) and \( > \)): Shade above the line.
  • Less than or equal to (\( \leq \) and \( < \)): Shade below the line.

For y \geq -\frac{3}{2}x - 3, shade the region above the line because it represents all solutions where y is greater than or equal to -\frac{3}{2}x - 3. Remember, since it's \geq (greater than or equal to), we include the line itself. So, use a solid line in your graph. The shaded area shows all the points that satisfy the inequality.
Testing Points
Testing points helps confirm which region to shade. To test if the chosen region is correct, pick a point that is not on the line. For our inequality, a good test point is
(0,0) because it's easy to calculate. Follow these steps to test:
  • Substitute the test point into the inequality: \[0 \geq -\frac{3}{2}(0) - 3 \].
  • Simplify to \[0 \geq -3 \].
  • Since this statement is true, the point (0, 0) is in the solution region.

If the inequality holds true for the test point, it means the region you chose to shade is correct.
It's a great way to double-check your work and make sure you've shaded the right area on the graph.

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Most popular questions from this chapter

Graph the linear inequality \(x-y \geq-2\)

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(y\) -axis, point (-6,2)

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(4 x+3 y=1\), containing point (0,0)

Graph the linear inequality \(y \geq-\frac{1}{3} x-2\)

Find the equation of each line. Write the equation in slope-intercept form. Containing the points (-2,0) and (-3,-2)
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